A discrete version of Koldobsky's slicing inequality

Abstract

Let $\#K$ be a number of integer lattice points contained in a set $K$. In this paper we prove that for each $d\in \mathbb{N}$ there exists a constant $C(d)$ depending on $d$ only, such that for any origin-symmetric convex body $K\subset {\mathbb{R}}^d$ containing $d$ linearly independent lattice points $$\#K\le C(d)\max (\#(K\cap H)){\mathrm{vol}}_d(K{)}^{\frac{d-m}{d}},$$ where the maximum is taken over all $m$-dimensional subspaces of ${\mathbb{R}}^d$. We also prove that $C(d)$ can be chosen asymptotically of order $O(1{)}^d{d}^{d-m}$. In addition, we show that if $K$ is an unconditional convex body then $C(d)$ can be chosen asymptotically of order $O(d{)}^{d-m}$.

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